geometry in nature fishy stuff...geometry in nature – dornach oct 2013 ©john blackwood, 93...

Geometry in Nature – Dornach Oct 2013

________ ____ ©John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 [email protected], Morphology.org

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Lecture three

FRI 11-10-2013, 9.00am to 10.30am

Geometry in Nature – Fishy stuff Three talks to Mathematics conference at the Goetheanum

by John Blackwood

The first presentation considered the ideas and natural expression of line, foci on the line,

rhythms in the line and ended with the question that asked “in what context was this all this

happening?”.

The second presentation hypothesised that all of the kingdoms were able to be seen as

having an underlying architecture and that this was an expression of a distinct tetrahedral

structure, that is a distinct transform of the tetrahedron and that was unique to the kingdom

considered.

The third presentation is to do only with one “form experiment” that I tried, related to

aspects of the fish form.

A summary of the third presentation follows:

• This talk attempts to cover:

• The notion of a leap for the forms allowing the merely living (plants) to active

mobile soul engagement (animalic).

• That from plant to animal there is again a 90 degree leap, from the vertical to

basically horizontal.

• That there is an evolutionary memory (as it were) of the plant in the spiralations of

the scales of the fish.

• Always a new start – all kingdoms go through revised repetitions of their earlier

incorporations.

•

• Animal form?

• I started this little exploration with the assumption that there was indeed a special

tetrahedron, of some sort – for the generally animalic, the sentient, the conscious –

however dim or alert.

• Further that it had to be a transformation of the tetrahedron for the plant world (or

was that back to front?).

• It seemed to be that there could be something took over the architecture and took the

plant spine and made it become horizontal.

•

• 90 degrees again!

• In terms of physically visible evolution we can imagine a

huge leap from plant form (mainly vertical) to animailc

gesture (basically horizontal).

• Even the patterns living in the living plant are revisited to an

extent in the skin patterns – the scales – of the fish.



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• But can we find a geometry that indicates a continuity between the two?

• And will it lead from radial symmetry to bilateral symmetry?

•

• Radial symmetry.

• Another significant change between these two kingdoms is the change in the basic

symmetry.

• The plant kingdom has a basically radial symmetry. Viewed from above along the

spine, the circle, or rotation, presents itself as the plants raw symmetry. Observe

tree forms from a helicopter!

• The background to this slide is a Google Earth image of the Botanical Gardens in

Sydney, Australia. Bottom left is the Sydney Conservatorium of Music.

• Bilateral symmetry

• The symmetry becomes bilateral in the animal

kingdom for, when viewed from the front, there is

usually a definite reflection symmetry manifest. Ask

this fish.

•

• All bilateral …

• To get some of these images I had to resort to a trick.

Not being able to capture stills, short video clips were

taken.

• Then it was relatively easy to frame the direct front

(or back) views of the fish as they turned.

•

• Getting a front view …

•

• Gotcha!

• Fish forms?

• If the scalar pattern of the fish was to be what I

thought it could be then there was surely the

necessity for the world of the path curves to be

structuring the forms. If Edwards is right it is

there in the plant world without a doubt. What

about in the the fish world though – do these

curves also reach into the animalic world? Are

they enough?



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• Could the basic fish form body be described by any path curve profile?

• To begin with I simply looked at the profile as seen from the side of the animal.

•

• Choosing a fish …

• This exploration took me to the Fish Markets of Sydney – quite an institution that I

normally did not have cause to visit.

• I had no real idea where to start but I soon noticed that many fish only had small

scales.

• Also that the lateral line, a feature I quickly became aware of, was not very straight

in most cases.

• So I found myself looking for fishes with a more or less straight lateral line and

large(ish) scales.

•

• Scourge of the Parramatta River

• No one seemed to mind me busily taking photographs of their displays and stalls

either!

• To cut a long story short, I found an ideal feral fish, the dreaded carp. It was,

nevertheless, a beautiful animal even in death. A golden body covered in the most

amazing scales with very clearly defined markings along the lateral line points –

which could hardly be straighter.

•

• Starting points

• I had now to build a picture of all the criteria the fish form would need to fulfill and

developed a whole string of assumptions that I needed to allow myself.

• I did not know if any of these assumptions was true but one has to start somewhere

… so here goes ….

•

• Some assumptions

• I assumed that the fish lay horizontally (most of the time – although the pipe fish

made me wonder!!) in its environment. Here was the linear, the line like. These

particular pipe fish were like gymnasts!

• I assumed that the spine was reflected in the lateral line of the fish – a feature I

knew nothing of before this study.

• I assumed that the fish form would have two end points and that these were on the

spinal line – one near the head and the other

near the tail (but before it began to flair out).

•

• What was the fishy tetrahedron?

• That the tetrahedron for this complex form

would have the spine as one of the lines and that

its skew mate would be orthogonal to this spine

and also that it would be local (not infinitely far

away as with the plant).

• One of my first sketches is in the background

…. this crude drawing was done about mid

2009.

•

• A tetrahedron of the third kind?

• At first I assumed that the fishy/animalic



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tetrahedron lost the circling measure in the top line, as I presumed that the two

points would now coincide (big mistake).

• This would give the special case of step measure, in the top, line while retaining a

growth measure in the lower line (of the spine).

• I worked with this basic tetrahedron - initially.

•

• Could it be like this?

•

• …. and with associated fields in the front and back planes • The path curves turn out to be bilateral and as if they might describe a fish front

profile. Or so I thought.

• But, try as I might, I could not make them work.

• These curves (in the plane) are given by Edwards in his Projective Geometry, 1985,

p218,fig 154.

•

• Another model (3rd

kind?)

• So I thought to make a model (again!).

• • Doubts

• By this time I had my doubts as to whether this architecture could

work at all.

• So I had another closer look at the fish (never ignore the phenomena!).

• And lo!

•

• Koi (carp)

• The curves cut right across dorsal and ventral lines – and did

not try to avoid top and bottom edges. So they must be some

kind of continuous spiral. Even if the fins intruded through

this skin.

•

• Abandoned model …

• Hence this form of the model was abandoned, but I did now

have a conceptual model to try to work with. (The model ended up deteriorating in

the garden …)

• For the measure or rhythm in the top line had to be a circling measure, rather than a

step measure.

• I mention all this as it is important to see that it is not good to presume that what one

initially thinks will fit the case will necessarily actually fit!

•

• Then I assumed that:

• That the fish profile, from the side, would match up with a single planar path curve

and that curve was able to be given by the invariant triangle.

• That the spinal nodes were an approximation (in the middle) to a growth measure.

• That the front (and hence rear) views of the fish were of a bilateral symmetry.

• That this front profile was of the form of an ellipse (or close – for some fish were

obviously egg shaped).

• That the spiraling curves on the fish body formed by the nodes of the scale positions

are three dimensional path curves.



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• That these spirals (both clockwise and anti clockwise) would not be regular

logarithmic spirals but a spiroid form of some sort (that is an asymmetric spiral).

•

• Would this architecture work for the carp?

•

• Testing, testing …

• Every one of the assumptions made had gradually to be tested – at least to a certain

extent – and for this one sample of the carp.

• For if it didn‟t sort of work here on just one beautiful fish, then there was little point

in pursuing the hypothesis. It only had to work once – to enable a start to be made

on further work on the theory.

•

• Thawing out somewhat.

• The first thing was to take orthographic photographs of the beast – thus to establish

some concrete data.

• This involved a lot of fiddling and setting up – I was not in the habit of

photographing dead fish. And I somehow did all this without my wife knowing at

all. I was a bit nervous how she would respond to this large creature in the freezer!

In the freezer the body had bent somewhat and I had to allow it to unfreeze

sufficiently to be able to straighten it out.

• And after messing around taking pictures in the hot sun in the backyard we could

hardly eat it ...

•

• Orthographic views – early attempt

• These were my starting data …

• as an engineering draftsman …

•

• Does the invariant triangle

fit?

• Now I had to check the side

profile to see if it could even

remotely respond to a path

curve analysis.

• This took quite a few trials.

• A number of the assumptions

came under scrutiny here.

• (see background).



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• Body envelope

within an invariant triangle

• I had to assume the position of the spine was aligned with

the lateral line.

• I had to assume the positions of the end points or foci (near

head and near tail).

• I had to assume a position for the orthogonal top line (and

was initially unsure whether it should even be above or

below the creature) and marked as point P3/P4 on the

previous sketch.

• idea of the growth measure (also assumed) along the spine.

• I eventually found an envelope in which I believed the fish

body would be contained (except head and tail of course).

•

• Some kind of invariant triangle …

• What this was attempting to say was that the skin of the fish body profile was a tiny

part of an entire path field spanning all space.

• And that it was the same kind of field as the invariant triangle in the plane – as

shown in the second talk – but assuming (!), in this one case, an isosceles triangle.

• So many assumptions …. !

•

• Body cross section

• The next step was to see what the front profile could be approximated to.

•

• Ellipse section – front view

• This front view resolved itself into an approximate ellipse

with a major axis of 76 mm and a minor axis of 48 mm.

• This would have to be the maximum cross sectional area.

•

• Evolutionary transforms?

• As an aside, I wondered whether the animals cross

section would give a clue as to its chronological

incorporation into our physical world. Fish sections vary

a lot. But there may be a morphological sequence here

suggesting a precise chronology (that is, non fortuitous).

•

• Circle, ellipse, egg and …

• Early fish had, I understand, a

rounder simple cross section –

even circular.

• This carp appeared nearly elliptical.

• A further fish I studied appeared as an egg form profile –

and a good one too.

• Then there were other fish with re-entrant aspects, the

cow fish.

• Beyond Projective?



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• Was this last a geometry that belonged to something further than projective

geometry as such and if so what was it?

• This was a big question for me … as I could not see the morphology of the sentient

creatures being limited to the projective, wonderful though this was.

• Here I hit a conceptual brick wall –

I knew I did not know enough

math!

•

• Reverse engineer? • Having now found two profiles to

work with I thought to attempt a

“reverse engineering” exercise to

see if I could find a field of form

(Edward‟s term) that married with

the fish itself.

• My first rough layout sketch is

shown here …

•

• A more detailed improvement …

•

• What kind of surface spirals?

• The next step was to see if I could find spiral forms in the planes through head and

tail areas, and the top line – which might generate the spiral curves on the skin

surface.

• It was obvious that these forms could not be as straight forward as the plant forms

regular logarithmic spirals.



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• But what were they?

•

• Special spirals – spiroids

• What was the „spiral‟ shape I was looking for? I

thought it might be what I came to call a “spiroid”, that

is, “something like a spiral” – but not an equiangular

spiral as in plant form fields. Now to test it.

•

• …Projecting scale points

• The scale points were projected on to

the front plane that slanted down from

the upper line through the front of the

fish (red dots highlighted with the

purple triangles).

• Now the question was could my

spiroid form concept match this

empirical curve?

•

• Failure again!

• Try as I might I could not get a

reasonable match.

• None of the spiroids (red circles) in the

envelope of spiroids would pass

through my red data points (red dots).

•

• Multiple trials (2009)

• I must have tried a dozen times …..

•

• Another idea …

• Nothing worked even half reasonably.

• What was I missing?

• Dumbo. Then it occurred to me that the circling measure in the top line did not have

to be based on rays from the point at equal angles.

• I had in fact been employing a very special case, and assuming it had to fit. But it

would not.

•

• Spiroid field (red curves)

• The angles about the point did not have

to be equal. And this was a case that

Edwards had shown me years and years

before and which had given a quite

pretty picture.

• At that time I had no idea that this

beautiful picture might actually be

complicit in some natural architecture.

•

• Original spiroid drawing – sometime

last millenia



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• This was a

case of a

circling

measure of

points (in a

line) and a

circling

measure of

lines (in a

point) but

where the

point was not

related

specially to

the center of

the circling

measure of

points (if that

makes sense).

•

• 1/11/2009

• So I tried this kind of spiroid – except that now centers of

the circling measures were limited to being perpendicular

to each other.

• Still this was something of a special case too … but

allowing for a reflection symmetry.

•

• Symmetrical spiroid

• Now I took only a few data points at significant places –

at top, bottom and on (approximately) either side. My

new spiroid would have to fit at least these few if the hypothesis was to be vaguely

valid!

• The drawings were getting larger all the time!

• (Take complete original to Dornach?).

• If I remember rightly this worked first time!

•

• Back plane too?

• If this worked then so should the projections on to the back plane. Did they?

• If they did then it could mean that this funny peculiar but much more general spiroid

may find a place in front and back planes and so help with understanding the skin

surface of a ubiquitous natural form – the fish.

• Those years ago I would never have even dreamt it could actually apply to the “real

world”!

•

• Subsets of subsets

• This structure has a similarity to the plants basic field structure – but is far more

general … encompassing a myriad more form fields due to the flexibility the fields

and curves now have.



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• This would mean that the set of forms that was applicable to the fish forms

contained, as a subset, the forms of the plant world, just as the mineral forms

seemed to be a subset of the plant forms.

•

• Intersecting spiroid cones … 1/11/2009

• The skin surface as the resultant of two intersecting spiroid cones.

•

•

• Forms through tangents …

• … and tangents are lines. Our thesis is still

within the initial brief – which asserted the

primacy of the line.

•

• Form through tangent points

• Needless to say such a curve can be plotted via

its lines and points and planes.

• This drawing (background) shows the point-

wise construction.



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• I did not get to the plane-wise – which would be difficult anyway.

•

• A cognitive picture

• I was now closer to an expression of – or so I thought – the cognitive picture as it is

expressed in Steiner‟s early philosophical/ spiritual work, i.e. his Philosophy of

Freedom.

• For this says, to my understanding, that reality meets us when we truly unite concept

and percept.

• But this only happens through a constant weaving between the two worlds – which

are initially separated for our current

consciousness.

• This practice has to be the new Yoga, for it is

a „breathing‟ between two kinds of Maya –

leading to an eventual resolution of the two

initial illusory experiences. Is this the

transformed Magi

and Shepherd

paths?

•

•

• Concept +

Percept =

Reality!

• Are we getting

there … even just

a little way

towards it?

•

• 3/11/2009 … state

of the art!

• Is this then the

fishy tetrahedron?

• The curve shown



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would be just one of thousands that sculpt the skin surface of the fish.

• These curves would also be part of the infinite field which surrounds the fish and in

which it is embedded (so to say) and – dare one suggest – sustained and created in

the first place.

• The fishes formative field would not be derivative – but the fish body itself would

be – and embedded in its field.

•

• Fishy field of form … !?

•

•

• “Tetrahedron of the third kind”?

• Was this then the tetrahedron “of the third kind”?

• Or at least was it at an early stage in its evolution?

•

• A single sample!

• But this was only one curve on one surface of one fish of one species.

• So many more questions arose …

• How true to the whole form was this skin field – how did it fade into head and limb

structures, abrubtly or seemlessly or what?

• What of other species of fish?

• How to include the evident curvature in many if not most of the lateral lines in the

fish population of the world? This strongly suggested there was a further and deeper

step yet to be taken. However I thought to try other fish species.

•

• Mullet

• So it was back to the fish markets!

• Again I sought a fish with an inherent

straightness and reasonably sized scales.

• I found the sleek Sea Mullet!

•



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• No lateral!

• I did the photography that would give me the orthographic views of this animal.

• It quickly became clear that this creature did not even have a lateral line! I cooked it,

partially skinned it – no sign of a lateral line. It still had a spine of course.

• Then I looked in the books. And I find stated that here was a species with no lateral

line. I still thought an analysis was worth pursuing. So I attempted to get the

orthographic views I needed …

•

• Mullet – orthographic views

• This was as far as I had got to….

•

• Trout

• Then came an offer from Simon Charter in England, as I had mentioned the trout as

a possible candidate.

• He sent me some exquisite trout photos.

• These I attempted to analyse …

•

• A thing of beauty …

•

•

•

•

•

•

• Lateral line



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• •

• An analysis …

• This is where we had got to about this time last year. But more data was needed …

at least the trout had respectable and identifiable lateral line!

•

• Mystery of sentient form

• This seemed to be only a tiny step into the outer morphology of the sentient

creature.



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• Any full animalic tetrahedron would have to include the curvatures in the spine –

this was clear from many a fish species, even if the carp (and triut) lateral line was

largely straight.

• And what would bring about the head, rhythmic and tail form geometries – which

had well and truly begun to intervene with the very earliest of fish forms?

• For me this was a big research question – how does soul intervene morphologically?

•

• Line – or curve?

• If spines could curve – was the line, the straight line, the fundament I thought it

was?

• Was the curve the thing? Strings anyone?!

• Were spines really macro strings?

• What raw structure would give the basis for bird, mammal, reptile among many

others?

•

• And then there is the human…!

• And another leap of 90 degrees ….

• What is the meaning of this insistent and repeated orthogonality? What other major

step allows the human spirit entry into an earthly form?

•

•

•

•

•

geometry in nature fishy stuff...geometry in nature – dornach oct 2013 ©john blackwood, 93...

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